Public blockchains use mining as a means to decide who gets to add the next block. That is all it is good for. That is useful in public chains because 1 entity adding multiple blocks in a row basically gives them control of the history.

In a private chain where you know all the participants this doesn't matter since there are easier ways to decide who gets to go next(ie. round robin). So no mining is needed for a private chain and it is no less secure(probably even more secure than most public chains due to the cheapness of 51% attacks )